I am struggling with following problem. Consider 2A class in Co1 having 819*759*75 elements. Each element a from 2A have two representatives in Co0. Element a corresponds to E8 sublattice in Leech defined as {v: av=-v} where I call by a also proper preimage in Co0. Now the opposite having E8 sublattice L in Leech I can find element a(L) in 2A class.

My goal is to find relation between Order(ab) for a,b in 2A and corresponding geometry of two E8 sublattices. The Order(ab) can be 2,3,4,5,6.

Take any other sporadic group g and certain conjugacy class cg of order 2 elements. Is it known possible values of Order(ab) for a,b in cg ? Can this help to connect some "lattice" to the group g ?

Regarding your last question: McKay observed that the 2A elements in the Monster, Baby Monster, and Fischer24 yield affine $E_8$, $E_7$ and $E_6$ diagrams of conjugacy classes, where the magic numbers on nodes are the orders of the products. There are some partial explanations in the literature (search for Yamauchi, Lam, and collaborators) using Ising vectors in vertex operator algebras.
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S. Carnahan♦Feb 9 '12 at 5:37

Thank you for this answer ! Can you explain a little what are "magic numbers on nodes" ?
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Marek MitrosFeb 9 '12 at 8:48